Prediction errors and the winner’s curse.∗

Olivier Compte†

September 2004

Abstract

A popular explanation for the winner’s curse is that in correlated or interdepen-

dent value environments, bidders fail to take into account the information on other’s

estimate (hence, given the correlation or the interdependence, on own valuation)

conveyed by winning.

Another explanation, initially proposed by Capen and al. (1971), is that bidders

make estimation errors, and that competition induces a selection bias in favor of

most optimistic bidders.

The main purpose of this paper is to show that these explanations are not

equivalent. In particular, the latter one extends to settings in which values and

estimates are drawn from independent distributions, while the first does not.

The paper also discusses the role of over-confidence in the accuracy of own

signals in explaining the winner’s curse and its persistence: we find that both

explanations build on errors that may have their roots in that same cognitive bias.

Key words: Winner’s curse, private values, prediction errors, overconfidence.

∗This paper builds on Section 2 of ”The winner’s curse with independent private values”. I thankPhilippe Jehiel, Eric Maskin, Ran Spriegler, Jean Tirole, Shmuel Zamir as well as seminar participantsat CORE, ESSET 2001 (Gerzensee), the Institute for Advanced Studies (Princeton), the University ofPennsylvania, Université d’Aix-Marseille (Greqam) for helpful comments.

†CERAS-ENPC, e-mail compte@enpc.fr.

1. Introduction

In some competitive environments, “successful” bidders are not so successful after all:

“successful” bidders (that is, those who won the competition) tend to obtain returns

that (on average) lie below initial projections. This discrepancy between realized and

anticipated returns, and the possibility that winning bidders end up making losses, has

been called the winner’s curse.1

The most popular explanation for the winner’s curse is based on bidders’ failure to

incorporate the information conveyed by winning into their bidding strategy. Winning

typically conveys information on others’ value estimates. Thus, when valuations are

interdependent or drawn from correlated distributions, winning conveys information on

one’s own valuation. If bidders fail to take into account this information, they end up

with a biased estimation of their valuation.

Another explanation, initially proposed by Capen and al. (1971) in the context of

competition for oil fields, is that bidders make estimation errors, and that competition

induces a selection bias in favor of most optimistic bidders. Capen and al. (1971)

consider the problem of bidding for a tract that has the same (unknown) value for each

bidder, and they argue that:

“in competitive bidding, the winner tends to be the player who most over-estimates true tract value.... [So] a player tends to win a biased set of tracts- namely those on which he has over estimated value or reserves”.

Do these explanations differ, or are they just two different ways to say the same

thing?

Looking at the many textbooks and academic papers on the topic, one could be

tempted to opt for the latter view. Indeed, what is most often retained from Capen

and al.’s example is that bidders face a common value environment, and that they fail

1For an introduction to winner’s curse, see for example Milgrom (1989) or Thaler (1992, chapter 5).For evidence of the winner’s curse in experiments, see Bazerman and Samuelson (1983) and Kagel andLevin (1986). See also Kagel and Levin (2002) for a recent survey of experimental evidence.

1

to incorporate the information conveyed by winning into their bidding strategy; hence

the curse.

This paper will argue that these explanations do differ. In a nutshell, we will show

that while the first explanation relies on the fact that bidder’s valuation are correlated

-or interdependent, Capen and al’s explanation does not.

Intuitively, the reason is that competition tends to select optimistic bidders, whether

bidders’ valuations are drawn from independent distributions or not. One could for

example re-write Capen and al.’s argument as follows:

In competitive bidding, a bidder is more likely to win when he over-estimateshis own value for the object. So a bidder tends to win a biased set of objects- namely those on which he over-estimates his own value.

The next Section proposes a model that will formalize that intuition. The main

feature of our model is that it allows for prediction or estimation errors, which may for

example result from players being overconfident in the accuracy of their own signals.

Because of these predictions errors, bidders may either make optimistic or pessimistic

predictions. The point is that even if on average these errors cancel out, competition

induces a selection bias in favor of optimistic bidders. We show in Section 3 that this

selection bias exists even when valuations are drawn from independent distributions,

hence even when winning conveys no information on one’s own valuation; and failing to

take into account this selection bias is potentially harmful to bidders.

Finally, in Section 4, we analyze a common value setting and characterize the two

possible sources of curse: the standard one, based on the agent’s failure to take into

account the information on other’s estimates conveyed by winning, and the second

one, based on the fact that competition tends to select optimistic bidders. These two

effects are compared in a simple example. We then discuss more generally the role of

overconfidence in own signals in explaining the winner’s curse and its persistence.

2

2. A model with prediction errors.

Our first objective is to present Capen and al.’s insight in the simplest way, and show

that this insight is valid in a private value setting. Our model builds on an example

given in Milgrom (1989).2

Consider the problem of bidding for construction contracts. There are n potential

contractors bidding for a construction contract. Each contractor i = 1, ..., n has a cost

Ci of doing the job. Contractors however do not know precisely what the job will cost,

but they get an estimate Xi of Ci.

The process by which contractor i gets the estimate Xi is not modelled. To an

outside observer however, the costs Ci and the estimates Xi can be regarded as random

variables. For simplicity, we assume that

Xi = Ci + eεi,where eεi is an estimation error. The errors eεi, i = 1, ..., n are assumed to be independentacross bidders, independent of costs and satisfy:

E[eεi] = 0.So the estimate Xi is unbiased on average.

In this Section and the next one, we focus on the private cost case, in which the

individual costs Ci are drawn from independent distributions. We also assume that the

distributions over costs and errors are non-degenerate. Formally, we assume that the

cost Ci and the error eεi each admit a density (denoted respectively fi and gi), that thesupport of each density is an interval, and that each density is positive on its support.

The symmetric case will refer to situations where the marginal distributions over Ci

are identical across bidders, and where the errors are drawn from identical distributions.3

We now turn to the main assumption of our model.

2Milgom (1989) considers a common value example, that we transform into a private value example.3That is, fi = f and gi = g.

3

In a standard Bayesian model, contractors would be assumed to know the joint

distributions over the costs Ci and estimates Xi. In such a model, each contractor i

would be able to compute, for each realization x of Xi, the conditional distribution over

costs given x, hence also the conditional expectation of costs

Y ∗i (x) ≡ E[Ci | Xi = x].

Here, in contrast, we shall assume that each agent is aware that costs are distributed

independently, but that, based onXi, each bidder i forms a possibly erroneous predictionbYi of the cost Ci.4 ,5 One particularly simple case to analyze is one where the agent takeshis estimate at face value, without realizing that he makes estimation errors:

Assumption 1. bYi ≡ Xi.One possible interpretation of Assumption 1, which echoes numerous work in psy-

chology on overconfidence, is that contractor i believes that the estimate Xi has more

predictive content than it really has.

Comment: Of course, other specifications are plausible. For example, lessextreme forms of overconfidence would consist in assuming that contractori (erroneously) believes that

Xi = Ci + λεi (2.1)

where λ ∈ [0, 1).6 The interpretation of λ is that, although contractor irealizes that his estimate is subject to errors, he downplays the magnitudeof his own errors (by a factor λ). In that case, the prediction of costs wouldbe: bYi ≡ Eλ[Ci | Xi],

4The prediction bYi may thus differ from the Bayesian prediction Y ∗i .5Alternatively, we could assume that each bidder forms a possibly erroneous belief (i.e. conditional

distribution) over his cost Ci given his observation Xi. The prediction bYi should then be thought of theexpected value of cost conditional on Xi, taken under that possibly erroneous conditional distribution.

6This type of overconfidence is analogous to the one that appears in the finance literature, in whichtraders are assumed to be overconfident in the informative content or accuracy of their signals. Seefor example De Long et al. (1991), Kyle and Wang (1997), Odean (1998), or Daniel, Hirshleifer andSubrahmanyam (1998). These papers also discuss the relevent literature in psychology.

4

where the superscript λ stands for the fact that the expectation is takenassuming that the joint distributions over Xi and Ci is characterized by(2.1).7

Another plausible assumption would be that bidders make computation er-rors in computing conditional expectations. One possible specification wouldthen be: bYi ≡ Y ∗i + ηi,

where ηi is an independent random variable with zero mean.8

Because contractor i makes prediction errors, he will sometimes be too optimistic

about his cost, meaning that bYi < Y ∗i (Xi),and he will sometimes be too pessimistic about his cost, meaning that

bYi > Y ∗i (Xi).The difference between the correct and the actual prediction is denoted Hi:

Hi(Xi, bYi) ≡ Y ∗i (Xi)− bYi,and, for any given realization of Xi and bYi, this difference will interpreted as the agent’soptimism. Note that under Assumption 1, bidder i is not optimistic on average, because

by assumption EXi = ECi, hence prediction errors cancel out:

EHi = 0.

7To fix ideas, assume that Ci and εi are drawn from normal distributions, respectively N(c0,σ2) andN(0, ν2). Then bYi(Xi) = µ(λ)Xi + (1− µ(λ))c0with µ(λ) = σ2/(σ2 + λ2ν2). So the smaller λ, the more weight contractor i puts on Xi relative to theprior information c0.

8Bidders may fail to correctly compute conditional expectations even when they are given the struc-ture of the model and know all prior distributions. This has been checked experimentally in recent andindependent work by Goere and Offerman (2002). Their interpretation for these errors is that biddersfall prey to the base rate fallacy, and they notice the possibility of a curse induced by these errors, whichthey call a new’s curse.

5

Finally, we assume that contractors participate in a second price auction.9 Each

contractor i simultaneously submits a bid bi: the winner is the bidder with the lowest

bid, and he gets paid a price equal to the second lowest bid for doing the job.

In the private value setting we analyze, it is a dominant strategy to bid one’s own

cost prediction bYi, so we have:bi(bYi) = bYi

To complete the description of the model, we define formally what we mean by

winner’s curse. We let ∆i denote the expected difference between realized and predicted

costs, conditional on winning, that is,

∆i = E[Ci − bYi | i wins]Definition 1. We shall say that contractor i is subject to the winner’s curse when

∆i > 0.

In other words, the winner’s curse refers to situations in which winners underestimate

costs: on average, winners are disappointed when the actual cost is realized. Note that

∆i can be rewritten as

∆i = E[Y∗i − bYi | i wins] = E[Hi | i wins],

hence∆i can also be interpreted as bidder i’s expected optimism, conditional on winning.

It thus follows immediately that in a standard Bayesian model there can be no winner’s

curse, since then, Hi(Xi, bYi) = 0 with probability one.9We focus on the second price auction, so that we need not worry about the bidder’s beliefs about

other bidders’ bids. However, our analysis can be easily extended to first price auctions, assuming forexample that in equilibrium, bidders know (or have learned) the distribution over their opponent’s bids.Under Assumption 1, bi(bYi) would solve,

bi(bYi) = argmax(bi − bYi)Pr(bi < b−i).

6

3. Main result.

In this Section, we show that under Assumption 1, bidder i is subject to a winner’s

curse (even though costs are drawn from independent distributions) and that bidders

may end up making losses when competition is fierce enough. We will also examine how

increased competition affects the curse.

3.1. The winner’s curse.

If bidder i were certain to obtain the contract, then on average, his prediction errors

would cancel out and he would not be subject to the winner’s curse. Formally, we would

have:

∆i = E[Hi | i wins] = EHi = 0.

When he competes with other bidders, bidder i is (typically) no longer certain to obtain

the contract, and he is more likely to win in events when he makes a low bid, hence

in events where his prediction is low. The consequence is that conditional on winning,

prediction errors do not cancel out anymore, because in these events where bidder i’s

prediction is low, bidder i also tends to be optimistic about the cost Ci (see Lemma

1 below). In other words, the auction induces a selection bias in favor of optimistic

bidders.

Formally, we will prove the following Proposition, which says that unless he wins

with probability one, bidder i is subject to the winner’s curse.

Proposition 1. Under Assumption 1, if 1 > Pr{i wins} > 0, then ∆i > 0.

Before proving Proposition 1, we show the following result, which says that condi-

tional on the event where bidder i makes a low prediction of cost (say, below a threshold

p), bidder i tends to be optimistic about his cost. Proposition 1 will then follow because

bidder i only wins in events where is prediction is below p = minj 6=i bYj , the smallestprediction made by the other players.

Lemma 1. Consider p such that 0 < Pr{bYi < p} < 1. Then, under Assumption 1, wehave: E[Hi | bYi < p] > 0.

7

Intuitively, for any cost realization, low realizations of the prediction bYi coincidewith low realizations of the error term eεi, hence the optimism.

Proof. Consider the random variable Zi = p−Ci. Under Assumption 1, bYi = Xi =Ci + eεi and E[Hi | bYi] = E[Ci − bYi | bYi], hence we have

E[Hi | bYi < p] = E[−eεi | eεi ≤ Zi].Since Eeεi = 0, and since εi is independent of costs, for any realization zi ∈ Zi that fallswithin the support of eεi, we have10

E[eεi | eεi ≤ Zi, Zi = zi] < 0. (3.1)

Since Pr{eεi ≤ Zi} ∈ (0, 1), the supports of eεi and Zi must overlap,11 hence we obtainthe desired inequality.

Proposition 1 is then obtained as an immediate corollary of Lemma 1:

Proof of Proposition 1: Define P = minj 6=i bYj . We have:∆i = E[Hi | bYi < P ] (3.2)

Since 1 > Pr{i wins} > 0, the support of P and bYi must overlap, and the result thenfollows from Lemma 1.12

Comment 1: Proposition 1 is closely connected to Capen and al.(1971)’s insight:

their result would correspond to the case where the distributions over costs are degen-

erate and concentrated on the same value for all bidders. Proposition 1 makes clear

that their insight is valid more generally, and that it does not rely on costs across bid-

ders being common or interdependent. In Capen and al., costs are common, but this

10 If the error term was not independent of cost, inequality (3.1) would still hold under the assumptionthat conditional on cost, the error term is unbiased, i.e. E[eεi | Ci = c] = 0 for all cost realizations c.11This is because each support is an interval and because eεi and Zi both admit a density that is

everywhere positive on its support.12For realizations p of P that fall above the support of bYi, E[Hi | bYi < p] = EHi = 0.

8

is a mere consequence of their (implicit) assumption that the distribution over costs is

degenerate, combined with a symmetry assumption.

Comment 2: While Proposition 1 illustrates that competition induces a selection

bias in favor of optimistic bidders, Lemma 1 illustrates that a similar selection bias may

also obtain without competition, when contractor i is just given an option to contract at

a some price p (chosen as in Lemma 1). Assume that contractor i accepts the contract

whenever he expects positive profits, that is whenever p− bYi is positive. Then the lowerthe error term, the more likely he accepts the contract, and as a result, conditional on

accepting the contract, contractor i is too optimistic about his cost.13

One application of this observation concerns the winner’s curse in buyer-seller rela-

tionships.14 If, as in Akerlof (1970), valuations of the buyer and the seller are interde-

pendent, and if the buyer fails to realize this, he will be disappointed ex post by the

value of the object he bought. One corollary of Lemma 1 is that the same phenomenon

may arise when valuations are drawn from independent distributions and the buyer only

gets an imperfect estimate of his valuation. If he makes prediction errors, he will be

more likely to buy whenever his prediction is optimistic.

Comment 3: A recent insight due to van den Steen (2004) is that relative overop-

timism (i.e. the fact that an agent tends to have an optimistic perception of his own

prospect, relative to the way others view his own prospect) may stem from the com-

binations of two things: the fact that the agent makes estimation errors in evaluating

various alternatives, and the fact that there are various alternatives to choose from. In

a similar vein, a corollary of Proposition 1 is that the same combination of factors (esti-

mation errors and choice among various alternatives) generates optimism (not relative

to other’s perceptions, but relative to the true prospects). Indeed, assume there is a

single agent, and interpret the index i as one of the possible projects that the agent may

undertake. Choosing the project for which he has the lowest cost estimate is equivalent

13Lemma 1 is closely connected to Brown (1974)’s insight: When a firm makes estimation errorsin evaluating the value of a project, and undertakes it whenever the estimation is above a threshold,the evaluation ends up being too optimistic on average. As in Capen and al., the analysis of Browncorresponds to the case where the distribution of cost would be degenerate.14This winner’s curse in bilateral negotiations has been examined in experiments by Samuelson and

Bazerman (1985).

9

to selecting the agent’s which has the lowest cost estimate in the auction. Proposition 1

then says that whichever project the agent ends up selecting (optimally), his estimation

on that project will be too optimistic on average.

Comment 4: Proposition 1 easily generalizes to more general specifications of the

prediction function. In particular, it holds under the ”computation error specification”

mentioned earlier.15 It also hold whenever the two following properties are satisfied:

(i) EHi = 0, which means that on average prediction errors cancel out, and

(ii) E[Hi | bYi < p] is decreasing in p, which means that lower predictions correspondsto higher levels of optimism.16

3.2. Implications for profits.

We investigate the implication for profits in the symmetric case. When there are few

bidders, the effect on profit is not dramatic because the winner receives a payment

P = minj 6=i bYj which may be substantially higher than his own prediction (bYi). Whencompetition gets fierce however, the difference between the prediction and the payment

received vanishes, and because of the curse identified above, the winner eventually makes

losses. Formally, we have:

Proposition 2. Consider the symmetric case. If there are only two contractors com-

peting, expected profits are positive. With sufficiently many competitors, expected

profits become negative.

Under Assumption 1, P = minj 6=iXj , and expected profits can be written as

Πi = Pr{Xi < P}[E[P −Xi | Xi < P ]−∆i].

15Under the computation error specification, we have E[Hi | bYi < p] = E[−ηi | ηi + Y ∗i < p], whichagain is positive since ηi is an independent variable with zero mean.16These conditions for example hold for the case examined in footnote 7. Indeed, we have:

E[Hi | bYi] = −µ(λ)− µ(1)µ(λ)

(bYi − c0)So when λ < 1, contractors put more weight than they should on the estimate Xi, and as a result,E[Hi |bYi] is decreasing in bYi (hence a fortiori, E[Hi | bYi < p] is decreasing in p).

10

When competition gets fierce, the term E[P −Xi | Xi < P ] vanishes, so profits becomenegative because ∆i is positive and does not vanish.

With only two competitors, contractor 1 wins when C2 + ε2 > C1 + ε1 and then

makes a profit equal to C2 + ε2 − C1. So Π1 can be re-written as:

Π1 = Pr{1 wins}[E[C2 −C1 | C2 − C1 > ε1 − ε2] +E[ε2 | ε2 > C1 + ε1 − C2].

Since Eε2 = 0 and since EC2 = EC1, both terms on the right-hand side are positive,

hence Π1 is positive as well.

3.3. Comparative statics.

How does the curse change when competition increases? When competition increases,

low realizations of P = minj 6=i bYj are more likely, hence when bidder i wins, he tends tohave a lower prediction bYi. If a lower prediction implies stronger optimism, then morecompetition should imply higher optimism conditional on winning, hence a stronger

curse. Indeed, we assume below that E[Hi | bYi] is decreasing in bYi, which means thatbidder i is more optimistic when he has a lower prediction. We have:

Proposition 3. Assume E[Hi | bYi] is decreasing in bYi. Then ∆i increases with thenumber of bidders.

The proof is standard and relegated to the Appendix.

A similar conclusion holds for the effect of competition on welfare. If the correct

prediction Y ∗i is smaller for lower values of the estimate Xi, or more generally, if E[Y∗i |bYi] is increasing in bYi, then competition will increase welfare, despite the fact that the

winner gets more optimistic.

It is worth pointing out however that this monotonicity is not guaranteed in general.

To see why, consider a case where errors are correlated with costs and where estimation

errors are more pronounced for higher realizations of cost. Then more competition may

make more likely the selection of contractors who simultaneously get a large realization

11

of cost and a large (but negative) error term.17 In that case, competition may be

detrimental to welfare.18

4. Discussion.

We have so far focused on the case where costs and errors are drawn from independent

distributions, and we have found that Capen and al.’s explanation for the winner’s curse,

based on the selection of optimistic bidders, applied to that setting. This is in contrast

with the standard ”informational” explanation: when costs and estimates are drawn

from independent distributions, other bidders’ bids or estimates cannot help a bidder

improve his assessment of own cost; hence failing to take into account that information

cannot generate a curse for winners.

In this Section, we wish to pursue our comparison between the two explanations and

deal with the case of common costs, that is, the case where the individual cost Ci is the

same for all bidders. We shall denote by C the random variable defining this common

cost 19 and focus on the symmetric case.

We start with the standard ”informational” explanation, based on the failure of

bidders to recognize that other bidders’ bids convey information about the cost.

17Such a correlation could arise quite naturally. If completion of the contract requires various skills,those who master them (hence are presumably more cost efficient) should also be better at estimatingthe cost of completing the contract.18A simple example that illustrates this point is the following. Assume costs are i.i.d. on [1, 2], and

that Xi = Ci + kζi(Ci − 1), with ζi uniform on [−1, 1], and k ∈ (0, 2). [So the error term is equal tokζi(Ci − 1), it is correlated with cost, but since E[kζi(Ci − 1) | Ci] = 0, Proposition 1 applies - seefootnote 10].When k < 1, the support of Xi is [1, 2 + k], and as competition increase, winners tend to have a

realization of Xi closer to 1, which means higher welfare and vanishing optimism (so this is an exampleof a case in which Proposition 3 does not apply).When k > 1 however, the support of Xi is [2 − k, 2 + k], and as competition increase, winners tend

to have a realization of Xi closer to 2 − k, which means highest possible cost realization, and highestpossible optimism.19For any realization c of C, we have Ci = c for all i.

12

4.1. Case (a): The standard ”informational” explanation.

Following the popular explanation for the winner’s curse, we first consider the case

(labelled case a ) where each bidder i is assumed to correctly compute Y ∗i (Xi) = E[Ci |Xi], but to fail to recognize that other bidders’ estimates convey information about

his own cost (for example because he erroneously believes that costs are distributed

independently across bidders.20). Bidder i’s prediction is thus

bY ai (Xi) = Y ∗i (Xi),and it is independent of whether he wins or not. Define ∆ai as the expected difference

between realized and predicted costs, conditional on winning. That is, since we consider

the symmetric case,

∆ai ≡ E[C − bY ai | Xi ≤ Xj for all j].For any realization x of Xi, we have:

EhC − bY ai | Xi = x,Xi ≤ Xj for all ji = E [C | Xi = x,Xi ≤ Xj for all j]−E [C | Xi = x]

> 0,

implying that

∆ai > 0.

This inequality captures the standard “informational” explanation for the winner’s

curse, and ∆ai measures the size of the curse for bidder i. ∆ai is positive (hence bidder

i is too optimistic about his cost) because bidder i ignores the information conveyed

by winning. Unlike in the previous section however, optimism does not stem from a

selection bias in favor of optimistic bidders. If bidder i was not competing with other

bidders, his prediction bY ai (Xi) would be neither optimistic nor pessimistic. It wouldbe the correct one, given the information that he possesses. His prediction bY ai (Xi)becomes optimistic when he competes with other bidders and wins, because he (erro-

20This formalization corresponds to the fully cursed equilibrium examined in Eyster Rabin (2002).

13

neously) believes that winning conveys no information (and because winning is actually

bad news).

4.2. Case (b): The explanation based on selection of optimistic bidders.

Following the analysis of Section 3, we now consider the case (labelled case b ) where

bidders are assumed to take their estimate at face value, and make the following pre-

diction: bY bi (Xi) = Xi.We also assume that this prediction is independent of whether he wins or not, either be-

cause he has no doubt about his own cost being equal to Xi (extreme overconfidence), or

because, as in case (a), bidders erroneously believe that costs are drawn from indepen-

dent distributions. We will return to this assumption shortly (see subsection 4.4), and

discuss cases where bidders are less confident in their estimate and may be aware that

costs are identical across bidders (and thus attempt to draw inferences from winning).

On average, bidders’ prediction coincides with that of case (a):

E bY bi = E bY aiThe size of the curse however differs. Not only does bidder i ignore the information

conveyed by winning, but he also makes additional errors in predicting costs. These

errors exacerbate the curse, for the reason identified in Section 3. Defining

∆bi ≡ E[C − bY bi | Xi ≤ Xj for all j],we have:

∆bi −∆ai = E[Y ∗i (Xi)−Xi | Xi ≤ Xj for all j],

which by Lemma 1 is positive. The intuition is identical to that given before. Compared

to case (a), bidder i may be too optimistic about his cost (bY b < Y ∗i ) or too pessimistic(bY b > Y ∗i ). Because competition tends to select optimistic bidders, winners are even

more optimistic than under case (a), and ∆bi−∆ai captures the magnitude of that effect.

14

In summary, case (b) combines the two effects: the failure of bidders to recognize

that winning conveys information on cost (i.e. the “informational effect”), and the

endogenous selection of the more optimistic bidders (i.e. the “selection effect”).

4.3. Illustration.

To illustrate, we compare the magnitude of each effect in a simple example. Assume

that

C = c0 + γ and Xi = C + eεiwith γ and eεi drawn from independent distribution. Then we have

∆bi = E[−eεi | eεi < eεj ∀j] and ∆ai = E[C − Y ∗i (Xi) | eεi < eεj ∀j]We compare ∆ai and ∆

bi under two distinct assumptions:

(i) We start with the case where γ and eεi are drawn from identical distributions.

Then it is easy to check that:21

∆bi −∆ai = ∆ai .

Hence in that case, the informational effect and the selection effect both have the same

magnitude.

(ii) We now turn to the case where γ is concentrated around 0. 22 A bidder who

would know the distribution over cost and errors would realize that costs remain close

21To see why, observe that since γ and eεi are drawn from identical distributions, E[γ | Xi] = E[eεi | Xi],so

Y ∗i (Xi) = c0 +E[γ | Xi] = c

0/2 + [c0 +E[γ + eεi | Xi]]/2 = c0/2 +Xi/2

hence, since γ and eεi are drawn from independent distributions,

E[C − Y ∗i (Xi) | eεi < eεj ∀j] = c0 − c0/2− c0/2 +E[−eεi/2 | eεi < eεj ∀j]implying that

∆ai = ∆b

i/2

22This is the case examined by Capen and al.

15

to c0, whatever signal Xi is observed. Thus Y ∗i (Xi) ' c0, and

∆ai ' 0.

In contrast, ∆bi depends only on the distribution of errors, hence it remains unchanged

(and significant) even as γ gets concentrated around 0. Thus in that case, the win-

ner’s curse derives exclusively from the selection bias, and not from bidders ignoring

information embodied in others’ bids.

4.4. Over-confidence and the winner’s curse: further comments.

Throughout the paper, we have paid particular attention to the case where bidder i

makes prediction errors concerning his own cost. This corresponds to situations where

bidders understand that their estimation is subject to errors, but downplay the mag-

nitude of their own errors; the extreme case being one where bidders believe that they

are not making any estimation errors. We have found that this assumption alone could

be responsible for a winner’s curse in the private cost setting.

When we moved to the common cost setting, we have added the assumption that

bidders fail to recognize that they are facing a common value setting and that winning

means that they have the lowest cost estimate. Under this additional assumption, we

have found that whether bidders downplay the magnitude of their errors (case (b)) or

not (case (a)), they fall prey to the winners’ curse; and we could distinguish between

the standard “informational effect”, and what we called, the ”selection effect”.

In many settings however, bidders presumably do realize that they are facing a com-

mon value setting. This is certainly the case for example in the class room experiment

examined by Bazerman and Samuelson (1983), where students are asked to bid for a jar

filled with coins. In such settings, one key (empirical) issue is whether bidders would

modify their prediction after being told, say, the distribution over other bidders’ pre-

dictions.23 It would be quite reasonable to expect bidders to be influenced by others’

predictions, hence (unless they do not realize the connection between winning and the

23This is issue could be easily adressed in an experiment, but to our knowledge it has not beenadressed yet.

16

fact that others had a smaller prediction) to adjust their prediction of costs conditional

on winning. If the adjustment is strong, we should expect the winner’s curse to be

reduced substantially; and otherwise to persist.

What can be said concerning the magnitude of this adjustment?

First, in the extreme case where bidders believe that they are not making any errors,

information concerning other bidders’ estimates will not be used, hence the winner’s

curse will persist. More generally, the more bidders are over-confident in own signals

(i.e. the more they downplay the magnitude of their own errors), the more limited the

adjustment will be, and this should be true even if players have a correct idea of the

joint distribution over other players’ estimation errors.24

So over-confidence in own signals may be responsible for a winner’s curse in the

common cost setting as well, even when players do realize that they are facing a common

value setting, and even when they realize that winning conveys information on others’

estimates. But the winner’s curse now obtains through two possible channels: the

”selection effect”, because over-confidence in own signals induces bidders to overweigh

own estimate Xi; and the ”informational effect”, because over-confidence in own signals

also leads bidders to underweigh others’ estimates.

Of course, other reasons (besides over-confidence in own signals) may lead bidders

to disregard or put too little weight on others’ estimates: even when players have a

correct idea of the distribution of their own costs and estimations, and correctly compute

conditional expectations, they may have an inflated idea of the magnitude of errors made

by others or of the correlation of these errors,25 with the consequence that they fail to

incorporate in their prediction as much information as they should concerning other

24For example, in the case of normal distributions examined in footnote 3, if bidder i believe thatXi = C + λεi and Xj = C + εj , we have

E[C | Xi,Xj ] = µi(λ)Xi + µj(λ)Xj + (1− µi(λ)− µj(λ))c0

with µi(λ) =ν2

(λν)2+ν2+(λν2/σ)2and µj(λ) =

(λν)2

(λν)2+ν2+(λν2/σ)2. So the smaller λ, the more weight

contractor i puts on own signal Xi, and the lesser weight contractor i puts on contractor j’s signal Xj .25The experiment conducted by Bazerman and Samuelson (1983) actually reveals that errors may be

highly correlated and biased: the average prediction of students were 37% below the true value of thejar.

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bidders’ estimates.

Which type of error (downplaying own estimation errors or inflating others’) is more

prevalent in practice is an empirical question that would be worthwhile investigating in

the lab.

5. Conclusion

This paper has attempted to clarify the difference between two types of arguments

used in the literature to account for the winner’s curse. Both arguments are based

on mistakes made by the bidders. The first one is based on the failure of bidders to

(fully) incorporate in their predictions the information on others’ valuations conveyed

by winning. The second one is based on bidders not realizing that (or the extent to

which) their value estimate is subject to errors along with the fact that they are more

likely to win when their estimate is optimistic.

One implication of our analysis is that the winner’s curse phenomenon may be more

widespread and persistent than generally thought.

Indeed, the second argument applies very broadly, including to settings to which

the first argument does not apply (e.g. to settings where valuations are drawn from

independent distributions, or to common value settings where bidders share the same

data.26 ,27). In addition, a single and very common cognitive bias (i.e. over-confidence

in the accuracy of own signal) may be responsible for both mistakes. Finally, over-

confident bidders will continue to fall prey to the winner’s curse even if they are taught

that winning conveys information on others’ estimates.

Several issues remain to be addressed. (i) In theory, various types of errors may

explain the winner’s curse: failure to recognize that winning conveys information about

others’ estimate, over-confidence in own signals, under-confidence in the accuracy of

other’s signals, and testing the relevance of each explanation deserves further research.

26 In these settings, other bidders’ bids convey no additional information (because there are no addi-tional information to be obtained), hence the first argument does not apply.27One good illustration is the class room example mentioned earlier where students are asked to bid

for a jar filled with coins (See Bazerman and Samuleson 1983). They all share the same data (a view ofthe jar), and yet they fall prey to winner’s curse.

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(ii) We have portrayed the behavior of somewhat naive bidders who often take their

estimates at face value and bid accordingly, and never learn that their choice is sub-

optimal. In the long run, as experience builds up, we might expect bidders to become

aware that their choice is suboptimal (in particular when these choices lead to losses)

and adjust their bids accordingly. This is a line of research we have started in Compte

(2001).28

Appendix

Proof of Proposition 3: Let ϕi denote the distribution over Xi induced by fi and gi,

and Qi(x) = Pr{Xi > x}. Also let hi(x) = E(Hi | bYi = x). With n bidders, we have:∆(n)i =

Rhi(x)ψi(x)dxR

ψi(x)dx

where ψi(x) = ϕi(x)Qnj=1,j 6=iQj(x), and where the superscript (n) indicates the number

of bidders.

With one additional bidder, say bidder n+ 1, the bias ∆i becomes

∆(n+1)i =

Rhi(x)Qn+1(x)ψi(x)dxRQn+1(x)ψi(x)dx

Since both hi and Qn+1 are decreasing functions, and since the expectation of the

product of two decreasing functions is larger than the product of the expectations, we

have: Rhi(x)Qn+1(x)ψi(x)dxR

ψi(x)dx≥Rhi(x)ψi(x)dxR

ψi(x)dx×RQn+1(x)ψi(x)dxR

ψi(x)dx

which implies , ∆(n+1)i /∆(n)i ≥ 1.

28 In Compte (2001), we propose a model along this line, where bidders learn to set (optimally)a uniform mark-up on their cost estimate. In this model we find that private value and commonvalue settings both yield similar qualitative predictions concerning the effect of competition on biddingbehavior, namely, increased cautiousness.

19

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